平方根晶体与$B(\infty)$的平方根
Square root crystals and the square root of $B(\infty)$
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中文总结 AI 辅助
本文引入N-根晶体的通用幺半范畴，研究平方根$\mathfrak{gl}_n$-晶体，构造$B(\infty)$的平方根类似物并刻画其性质，还探讨其与经典晶体构造的差异。
中文摘要 AI 辅助
我们引入了一类通用的N-根晶体的幺半范畴，随后研究了平方根$\mathfrak{gl}_n$-晶体的特殊情形。这类对象包含Yu在半标准集值表格上的晶体。第一作者、Tong和Yu的前期工作表明，正则平方根$\mathfrak{gl}_n$-晶体可作为证明Grothendieck正性结果的有用工具。本文研究的对象超出正则情形，使我们能构造直极限晶体$B(\infty)$的平方根类似物。我们通过边缘大表格、Lusztig或PBW参数化、以及Nakashima-Zelevinsky多面体模型，给出了$B(\infty)$的平方根的若干刻画。我们证明该晶体具有简洁的特征标公式、展现非平凡的Demazure滤过，且在取适当张量积后可恢复Yu的半标准集值表格晶体。我们还探究了平方根晶体与经典晶体构造的诸多差异。
英文摘要
We introduce a general monoidal category of $\mathbf{N}$-root crystals and then study the special case of square root $\mathfrak{gl}_n$-crystals. The latter objects include Yu's crystals on semistandard set-valued tableaux. Prior work of the first author, Tong, and Yu showed that regular square root $\mathfrak{gl}_n$-crystals can be a useful tool for proving Grothendieck positivity results. The objects studied here go beyond the regular case and allow us to construct a square root analog of the direct limit crystal $B(\infty)$. We give several descriptions of our square root of $B(\infty)$, using marginally large tableaux, the Lusztig or PBW parameterization, and the Nakashima--Zelevinsky polyhedral model. We show that this crystal has a simple character formula, exhibits a nontrivial Demazure filtration, and recovers Yu's semistandard set-valued tableau crystals after taking appropriate tensor products. We also investigate a number of differences between square root crystals and classical crystal constructions.
